Optimal. Leaf size=82 \[ \frac {e \sqrt {d^2-e^2 x^2}}{d^2 x}-\frac {\sqrt {d^2-e^2 x^2}}{2 d x^2}-\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^2} \]
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Rubi [A] time = 0.08, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {850, 835, 807, 266, 63, 208} \[ \frac {e \sqrt {d^2-e^2 x^2}}{d^2 x}-\frac {\sqrt {d^2-e^2 x^2}}{2 d x^2}-\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 850
Rubi steps
\begin {align*} \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)} \, dx &=\int \frac {d-e x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{2 d x^2}-\frac {\int \frac {2 d^2 e-d e^2 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{2 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{2 d x^2}+\frac {e \sqrt {d^2-e^2 x^2}}{d^2 x}+\frac {e^2 \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{2 d x^2}+\frac {e \sqrt {d^2-e^2 x^2}}{d^2 x}+\frac {e^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{2 d x^2}+\frac {e \sqrt {d^2-e^2 x^2}}{d^2 x}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{2 d}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{2 d x^2}+\frac {e \sqrt {d^2-e^2 x^2}}{d^2 x}-\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 70, normalized size = 0.85 \[ -\frac {(d-2 e x) \sqrt {d^2-e^2 x^2}+e^2 x^2 \log \left (\sqrt {d^2-e^2 x^2}+d\right )-e^2 x^2 \log (x)}{2 d^2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 63, normalized size = 0.77 \[ \frac {e^{2} x^{2} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + \sqrt {-e^{2} x^{2} + d^{2}} {\left (2 \, e x - d\right )}}{2 \, d^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 254, normalized size = 3.10 \[ -\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}\, d}-\frac {e^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}\, d^{2}}+\frac {e^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, d^{2}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{3} x}{d^{4}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{2}}{2 d^{3}}-\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{2}}{d^{3}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e}{d^{4} x}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 d^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {d^2-e^2\,x^2}}{x^3\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{x^{3} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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